MEASURE CONTRACTION PROPERTIES OF CONTACT SUB-RIEMANNIAN …zelenko/meascontrhighfin.pdf · ....

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MEASURE CONTRACTION PROPERTIES OFCONTACT SUB-RIEMANNIAN MANIFOLDS WITH

SYMMETRY

PAUL W. Y. LEE, CHENGBO LI, AND IGOR ZELENKO

Abstract. Measure contraction properties are generalizations ofthe notion of Ricci curvature lower bounds in Riemannian geome-try to more general metric measure spaces. In this paper, we givesufficient conditions for a contact sub-Riemannian manifold witha one-parameter family of symmetries to satisfy these properties.Moreover, in the special case where the quotient of the contact sub-Riemannian manifold by the symmetries is Kahler, the sufficientconditions are defined by a combination of the holomorphic sec-tional curvature and the Ricci curvature of the quotient manifold.This generalizes the earlier work in [2] for the three dimensionalcase and in [14] for the Heisenberg group. To obtain our results weuse the intrinsic Jacobi equations along sub-Riemannian extremals,coming from the theory of canonical moving frames for curves inLagrangian Grassmannians [17, 18]. The crucial new tool here isa certain decoupling of the corresponding matrix Riccati equation.It is important to note that our measure contraction propertiesfor the considered class of sub-Riemannian structures cannot beimproved because the corresponding inequalities turn to be equal-ities for the corresponding homogeneous models. Using the samescheme, we also prove a version of Bonnet-Myer’s Theorem in oursetting.

1. Introduction

In recent years, there are lots of efforts in generalizing the notion ofRicci curvature lower bounds in Riemannian geometry and its conse-quences to more general metric measure spaces. One of them is the

Date: July 31, 2013.2010 Mathematics Subject Classification. 53C17, 53D10, 70G45, 34C10, 53C25,

53C55.Key words and phrases. Sub-Riemannian metrics, Measure Contraction proper-

ties, contact manifolds, Jacobi equations and Jacobi curves, Matrix Riccati equa-tion, Bonnet-Myers theorems.

The first author’s research was supported by the Research Grant Council of HongKong (RGC Ref. No. CUHK404512). The second author was supported in partby the National Natural Science Foundation of China (Grant No. 11201330).

1

2 PAUL W. Y. LEE, CHENGBO LI, AND IGOR ZELENKO

work of [20, 21, 26, 27] where the notion of curvature-dimension condi-tions was introduced. These conditions are generalizations of Ricci cur-vature lower bounds to length spaces equipped with a measure (lengthspaces are metric spaces on which the notion of geodesics is defined).In [24], it was shown that the curvature-dimension conditions coincidewith the pre-existing notion of Ricci curvature lower bounds in the caseof Finsler manifolds.

On the contrary, it was shown in [14] that the curvature-dimensionconditions defined using the theory of optimal transportation are notsatisfied on the Heisenberg group, the simplest sub-Riemannian man-ifold. (Note however that a type of curvature-dimension conditionswere defined in [7, 8] using a sub-Riemannian version of the Bochnerformula). It was also shown in [14] that the Heisenberg group satisfiesanother generalization of Ricci curvature lower bounds to length spacescalled measure contraction properties [26, 27, 23].

Measure contraction properties are essentially defined by the rate ofcontraction of volume along geodesics inspired by the classical Bishopvolume comparison theorem. In the Riemannian case, the measure con-traction property MCP(k, n) is equivalent to the conditions that theRicci curvature is bounded below by k and the dimension is boundedabove by n. In [14], it was shown that the left-invariant sub-Riemannianmetric on the Heisenberg group of dimension 2n + 1 satisfies the con-dition MCP(0, 2n+ 3). Such sub-Riemannian metric can be regardedas the flat one among all sub-Riemannian metrics on contact manifoldsof the same dimension.

The next natural task is to study the measure contraction propertyfor general (curved) sub-Riemannian metrics on contact manifolds and,in particular, to understand what differential invariants of such metricsare important for their measure contraction property. A natural wayof doing this is to analyse the Jacobi equation along a sub-Riemannianextremal. Usually in order to write the Jacobi equation intrinsicallyone needs first to construct a connection canonically associated with ageometric structure. The construction of such connection is known inseveral classical cases such as the Levi-Civita connection for Riemann-ian metrics, the Tanaka-Webster connection for a special class of sub-Riemannian contact metrics associated with CR structures ([28, 30]) orits generalization, the Tanno connection, to more general class of sub-Riemannian contact metrics associated with (non-integrable) almostCR structures ([29]). However, these constructions use specific proper-ties of the geometric structures under consideration and it is not clearhow to generalize them to more general sub-Riemannian structures.

MEASURE CONTRACTION PROPERTIES 3

An alternative approach for obtaining intrinsic Jacobi equation with-out preliminary construction of a canonical connection was initiated byA. Agrachev in [1] and further developed in [5, 17, 18]. In this approachone replaces the Jacobi equation along an extremal by a special, a pri-ori intrinsic curve of Langranian subspaces in a linear symplectic space,i.e. a curve in a Lagrangian Grassmannian. This curve is defined upto the natural action of the linear symplectic group. It contains allinformation about the space of Jacobi fields along the extremal andtherefore it is called the Jacobi curve of the extremal.

By analogy with the classical Frenet-Serret frame for a curve in anEuclidean space, for a curve in a Lagrangian Grassmannian satisfyingvery general assumptions one can construct a bundle of canonical mov-ing symplectic frames [17, 18]. The structure equations for these mov-ing frames can be considered as the intrinsic Jacobi equations, whilethe nontrivial entries in the matrices of these structure equations givethe invariants of the original geometric structures that can be used inprincipal for obtaining various comparison type results including themeasure contraction properties. Although the construction of these in-variants is algorithmic, to express them explicitly in terms of the origi-nal geometric structure is not an easy task already for sub-Riemanniancontact case ([2, 19]). Besides, in contrast to the Riemannian case, forthe proper sub-Riemannian structure the level sets of sub-RiemannianHamiltonian are not compact. Therefore, to control the bounds for thesymplectic invariants of the Jacobi curves along extremals additionalassumptions has to be imposed.

For the first time the scheme based on the geometry of Jacobi curveswas used in the study of the measure contraction properties in [2],where general three dimensional contact sub-Riemannian manifoldswere treated. Note that in this case there is a one-to-one correspon-dence between sub-Riemannian structures and three dimensional CRstructures with a distinguished contact form (or pseudo-hermitian struc-ture in the sense of Webster [30]). The canonical linear connection forthe latter structures (in an arbitrary odd dimension) was constructedby Tanaka and Webster ([28, 30]). If, in addition, the Reeb field of thedistinguished contact form is an infinitesimal symmetry of the struc-ture, then the structure also corresponds to a Sasakian metric. Formore details about the relations between all these structures see, forexample, [6].

Returning to the results of [2] it was shown there in particularthat the sub-Riemannian metrics associated with a three-dimensionalSasakian manifold satisfiesMCP(0, 5) if and only if the correspondingTanaka-Webster sectional curvature of the planes which are the fibers of


the contact distribution is bounded below by 0. Moreover, conditionscalled generalized measure contraction properties MCP(k; 2, 3) weredefined there. It was shown that a three-dimensional Sasakian mani-fold satisfies this condition if and only if the aforementioned Tanaka-Webster sectional curvature (of the fibers of the contact distribution)is bounded below by k.

In this paper, we generalize the results in [14, 2] to contact sub-Riemannian manifolds of arbitrary dimension having an infinitesimalsymmetry transversal to the given contact distribution, i.e. a vec-tor field transversal to the contact distribution which generates isome-tries of the sub-Riemannian structure. Making the quotient by thefoliation of integral curves of this vector field, one associates to suchsub-Riemannian manifold, at least locally, a Riemannian manifold (ofdimension one less than the original one), endowed with a symplecticform. If the (1, 1)-tensor relating the symplectic form with the Rie-mannian metric defines a complex structure, one has a Kahler structureon the quotient manifold and, equivalently, the sub-Riemannian metricon the original manifold corresponds to the Sasakian metric there.

We introduce new generalized measure contraction properties de-noted by MCP(k1, k2;N − 1, N) and discuss when a contact sub-Riemannian manifolds with symmetry satisfies them. If the afore-mentioned (1, 1)-tensor is parallel with respect to the Levi-Civita con-nection on the quotient Riemannian manifold, we give conditions forour generalized measure contraction properties to hold in terms of thebounds for this (1, 1)-tensor and for the Riemann curvature tensorof the quotient manifold (Theorem 2.1 below). In particular, in theKahlerian case the measure contraction property can be described interms of a lower bound for the holomorphic sectional curvature of theKahler structure and another lower bound for a certain expression in-volving the Ricci curvature and the holomorphic sectional curvature ofthe Kahler structure (Corollary 2.1 below).

To obtain these results we analyze the matrix Riccati equation as-sociated with the structure equation for the canonical moving frameof the Jacobi curves (Lemma 3.1 and section 4 below) on the basis ofcomparison theorems for matrix Riccati equations ([25, 16]) and use theexpressions for the symplectic invariants of the Jacobi curves (see sec-tion 5 below) based on the calculation in [19] adapted to our situationvia Theorem 8.10.

The key new observation that allows us to obtain the results is thepossibility of a certain decoupling of these equations after taking thetraces of appropriate blocks (as in the proof of Lemma 4.3 below): the


coupled equations after taking the traces yields to decoupled inequali-ties, which leads to the desired estimates, involving the Ricci curvatureof the quotient manifold. Surprisingly, the passage from equations toinequalities does not affect the sharpness of the estimates: our mea-sure contraction properties for the considered class of sub-Riemannianstructures cannot be improved because the corresponding inequalitiesturn to be equalities for the corresponding homogeneous models (seeCorollaries 2.2 and 2.3 below).

Note that in the Kahlerian case the Tanaka-Webster connection ofthe corresponding Sasakian metric on the original manifold can be in-terpreted as a natural lift of the Levi-Civita connection of the quotientRiemannian manifold ([10, Section 4]) and the estimates in Corollary2.1 can be easily reformulated in terms of the Tanaka-Webster cur-vature tensor of the Sasakian metric (see Remark 2.1 below for moredetail).

Besides, the method of the paper can be used when the (1, 1)-tensoron the quotient Riemannian manifold defines an almost Kahler struc-ture, because the lower bound for the symplectic invariant of Jacobicurves can be controlled as well (as follows from Theorem 5.7 below).

Finally, a similar scheme allows us to get a version of Bonnet-Myer’sTheorem for the upper bounds for the diameter of the sub-Riemannianmetrics under consideration (see Theorem 2.2 below). Note that thisresult improves the estimates for the diameter that can be deduced from[19] (see section 6, Corollary 5, items 5,6 in the Kalerian case there):the conditions for the sectional curvature of the quotient manifold thereare replaced by weaker conditions involving the Ricci curvature here.The decoupling of the Ricatti equation plays the crucial role in thisrefinement again.

Further note that there is a sub-Riemannian version of Bishop the-orem which was proved in [11, 3]. Unlike the Riemannian case, thisis very different from the generalized measure contraction propertiesMCP(k; 2, 3) (see [3]). This is essentially due to the fact that anyneighborhood contains points which are joined by more than one min-imizing sub-Riemannian geodesics.

Next, we state the condition MCP(k1, k2;N − 1, N) and some sim-ple consequences of the main results in more detail. Let M be a sub-Riemannian manifold (see Section 2 for a discussion on some basicnotions in sub-Riemannian geometry). For simplicity, we assume thatM satisfies the following property: given any point x0 in M , there is aset of Lebesgue measure zero such that any point outside the set is con-nected to x0 by a unique length minimizing sub-Riemannian geodesic.


By the result in [9], this is satisfied by all contact sub-Riemannianmanifolds.

Let t 7→ ϕt(x) be the unique geodesic starting from x and endingat x0. This defines a 1-parameter family of Borel maps. Let d be thesub-Riemannian distance and let µ be a Borel measure. The followingis the original measure contraction property studied in [26, 27, 23]:

A metric measure space (M,d, µ) satisfies MCP(0, N) if

µ(ϕt(U)) ≥ (1− t)Nµ(U)

for each point x0 and each Borel set U .Note that the condition MCP(0, N) implies the volume doubling

property of µ and a local Poincare inequality (see [26, 27, 23]). Bycombining this with the work of [12], this proves the Harnack inequalityand hence the Liouville property for the sub-Riemannian analogue ofharmonic functions.

Next, we introduce the new generalized measure contraction proper-tiesMCP(k1, k2;N − 1, N): A metric measure space (M,d, µ) satisfiesMCP(k1, k2;N − 1, N) if, for each point x0 and each Borel set U ,

µ(ϕt(U)) ≥∫U

(1− t)N+2M1(k2d2(x, x0), t)MN−3

2 (k1d2(x, x0), t)

M1(k2d2(x, x0), 0)MN−32 (k1d2(x, x0), 0)

dµ(x),

where

D(k, t) =√|k|(1− t),

M1(k, t) =

2−2 cos(D(k,t))−D(k,t) sin(D(k,t))

D(k,t)4if k > 0

112

if k = 02−2 cosh(D(k,t))+D(k,t) sinh(D(k,t))

D(k,t)4if k < 0,

M2(k, t) =

sin(D(k,t))D(k,t)

if k > 0

1 if k = 0sinh(D(k,t))D(k,t)

if k < 0.

Note, in particular, thatMCP(0, 0;N−1, N) is the same asMCP(0, N+2). If k1 ≥ 0 and k2 ≥ 0, thenMCP(k1, k2;N−1, N) impliesMCP(0, N+2). The reason for the notations in the conditions MCP(k1, k2;N −1, N) is clarified by Theorem 1.1 below.

Next, we state a simple consequence of the main results. For this,we let M be a contact sub-Riemannian manifold with symmetry andof dimension 2n + 1. This means that the sub-Riemannian struc-ture on M is defined by a contact distribution D and there is a sub-Riemannian isometry V0 which is transversal to D. We can extend thesub-Riemannian metric on M to a Riemannian one by the conditions


that the vector field V0 is of unit length and is orthogonal to D. Thecorresponding Riemannian measure is denoted by µ.

Let 〈·, ·〉 be the sub-Riemannian metric and let α be the contact formof D which satisfies the condition α(V0) = 1. Let J be the operatoron D defined by dα(v, w) = 〈Jv, w〉, where v and w are vectors in D.

Assume that the quotient M of M by the flow of V0 is a manifold. Then

〈·, ·〉 and J descend to a Riemannian metric and a operator on M , stilldenoted by 〈·, ·〉 and J , respectively. Finally, assume that (〈·, ·〉 , J)

defines a Kahler structure on M .

Theorem 1.1. Assume that the Riemann curvature tensor Rm of the

Kahler manifold M satisfies

〈Rm(Jv, v)v, Jv〉 ≥ k1|v|4

and

Rc(v, v)− 1

|v|2〈Rm(Jv, v)v, Jv〉 ≥ (2n− 2)k2|v|2.

for all tangent vectors v. Then the metric measure space (M,d, µ) sat-isfies MCP(k1, k2; 2n, 2n+ 1), where d is the sub-Riemannian distanceof M .

Remark that the first condition in Theorem 1.1 says that the holo-

morphic sectional curvature of M is bounded below by k1. As a corol-lary, we have the following result of [14] mentioned above as a specialcase.

Theorem 1.2. [14] The Heisenberg group of dimension n equippedwith the standard sub-Riemannian distance d and the Lebesgue measureµ satisfies MCP(0, 0; 2n, 2n+ 1) =MCP(0, 2n+ 3).

The complex Hopf fibration U(1) → M = S2n+1 → M = CPn canbe equipped with a sub-Riemannian metric such that M becomes acontact sub-Riemannian manifold with symmetry. In this case, V0 isthe infinitesimal generator of U(1)-action and the induced Riemannian

metric on M is the Fubini-Study metric (see [22]).

Theorem 1.3. The complex Hopf fibration equipped with the abovesub-Riemannian distance d and the measure µ satisfies the conditionMCP(4, 1; 2n, 2n+ 1). In particular, it satisfies MCP(0, 2n+ 3).

We also remark that the estimates for the proof of Theorem 1.2 and1.3 are sharp (see Corollary 2.1, 2.2, and 2.3 for more detail).

Finally, we also prove a Bonnet-Myer’s type theorem in our setting.


Theorem 1.4. Assume that the Riemann curvature tensor Rm of the

Kahler manifold M satisfies


for some positive constant k1. Then the diameter of the manifold Mwith respect to the corresponding sub-Riemannian metric is less thanor equal to 2π√

k1.

On the other hand, if

Rc(v, v)− 1

|v|2〈Rm(Jv, v)v, Jv〉 ≥ (2n− 2)k2|v|2,

for some positive constant k2, then the diameter of the manifold Mwith respect to the corresponding sub-Riemannian metric is less thanor equal to π√

k2.

In the next section, some basic notions in sub-Riemannian geometrywill be recalled and the main results of the paper will be stated. Therest of the sections will be devoted to the proof of the main results.

2. The Main Results

In this section, we recall various notions in sub-Riemannian geometrywhich are needed and state the main results of this paper. A sub-Riemannian manifold is a triple (M,D, 〈·, ·〉), where M is a manifold ofdimension n, D is a sub-bundle of the tangent bundle TM , and 〈·, ·〉 isa smoothly varying inner product defined on D. The sub-bundle D andthe inner product 〈·, ·〉 are commonly known as a distribution and asub-Riemannian metric, respectively. A curve γ(·) is horizontal if γ(t)is contained in D for almost every t. The length l(γ) of a horizontalcurve γ can be defined as in the Riemannian case:

l(γ) =

∫ 1

0

|γ(t)|dt.

Assume that the distribution D satisfies the following bracket gener-ating or Hormander condition: the sections of D and their iterated Liebrackets span each tangent space. Under this assumption and that themanifold M is connected, Chow-Rashevskii Theorem (see [22]) guar-antees that any two given points on the manifold M can be connectedby a horizontal curve. Therefore, we can define the sub-Riemanniandistance d as

(2.1) d(x0, x1) = infγ∈Γ

l(γ),


where the infimum is taken over the set Γ of all horizontal pathsγ : [0, 1]→M which connect x0 with x1: γ(0) = x0 and γ(1) = x1. Theminimizers of (2.1) are called length minimizing geodesics (or simplygeodesics). As in the Riemannian case, reparametrizations of a geo-desic is also a geodesic. Therefore, we assume that all geodesics haveconstant speed.

In this paper, we will focus on contact sub-Riemannian manifoldsmeaning that the distribution D is given by the kernel of a 1-form α,called a contact form, defined by the condition that the restriction of dαto D is non-degenerate. We also assume that there is sub-Riemannianisometry V0 on M which is transversal to D. The sub-Riemannianisometry V0 is a vector field for which the flow preserves the distribu-tion D and the length of horizontal vectors. For simplicity, we willalso assume that the quotient of M by the flow of V0 is a manifold

denoted by M . Let µ be the volume form on M defined by the condi-tion µ(V0, v1, ..., vn−1) = 1, where v1, ..., vn−1 is any orthonormal familyof horizontal vectors. The measure corresponding to µ will also bedenoted by the same symbol.

Since the distribution is contact, the sub-Riemannian distance x 7→d(x, x0) is locally semi-concave on M − {x0} by the result of [9]. Inparticular, the sub-Riemannian distance is differentiable almost every-where. Therefore, outside a set of measure zero, the points can beconnected to x0 by a unique length minimizing sub-Riemannian geo-desic. This also defines a family of Borel maps ϕt : M →M such thatt 7→ ϕt(x) is the unique length minimizing geodesic connecting x andx0.

Let 〈·, ·〉 and |·| be the sub-Riemannian metric and the correspondingnorm. Recall that the restriction of dα to the distribution D is non-degenerate. This defines an invertible endomorphism J : D → D bydα(v, w) = 〈Jv, w〉, where v and w are contained in D. Both the sub-

Riemannian metric and the operator J descend to the manifold M .The corresponding Riemannian metric and operator are denoted bythe same symbols.

Theorem 2.1. Assume that the tensor J is parallel and the Riemanncurvature tensor Rm satisfies

〈Rm(Jv, v)v, Jv〉 ≥ k1|v|2|Jv|2

and

Rc(v, v)− 1

|Jv|2〈Rm(Jv, v)v, Jv〉 ≥ (2n− 2)k2|v|2.


Then, for any Borel set U ,

µ(ϕt(U)) ≥ (1− t)2n+3

λ1λ2

∫U

M1(k1(x), t)M2n−22 (k2(x), t)

M1(k1(x), 0)M2n−22 (k2(x), 0)

dµ(x),

where

f(x) = d(x, x0),

k1(x) = f(x)2

(k1 +

(V0f)2(x)

λ22

),

k2(x) = f(x)2

(k2 −

(V0f)2(x)

4(2n− 2)

(λ3 + 2λ2

1

)),

λ1 and λ2 are upper bounds of the

operator norms of J and J−1, respectively,

λ3 is an upper bound of tr(J2).

Therefore, under the above assumption, the metric measure space(M,d, µ) satisfies MCP(k1, k2; 2n, 2n + 1). In particular, if k1 and k2

are non-negative, then it satisfies MCP(0, 2n+ 3).

If, in addition to the assumptions of Theorem 2.1, we assume that

the operator J satisfies J2 = −I (i.e. the manifold M is Kahler), thenTheorem 2.1 specializes to

Corollary 2.1. Assume that (M, 〈·, ·〉 , J) defines a Kahler manifoldand that the Riemann curvature tensor Rm satisfies


and

Rc(v, v)− 1

|v|2〈Rm(Jv, v)v, v〉 ≥ (2n− 2)k2|v|2.

Then

µ(ϕt(U)) ≥∫U

(1− t)2n+3M1(k1(x), t)M2n−22 (k2(x), t)

M1(k1(x), 0)M2n−22 (k2(x), 0)

dµ(x)

for any Borel set U , where

k1(x) = f(x)2(k1 + (V0f(x))2), k2(x) = f(x)2

(k2 +

1

4(V0f(x))2

).

In particular, the metric measure space (M,d, µ) satisfies the conditionMCP(k1, k2; 2n, 2n+ 1).


Corollary 2.1 is sharp in the sense that all the inequalities in Corol-lary 2.1, including both assumptions and conclusions, are equality inthe case of the Heisenberg group and the complex Hopf fibration.More precisely, the Heisenberg group is the sub-Riemannian manifoldwhere M is the 2n + 1-dimensional Euclidean space with coordinatesx1, ..., xn, y1, ..., yn, z. The distribution D is given by the span of thevector fields {

∂xi −1

2yi∂z, ∂yi +

1

2xi∂z

∣∣∣i = 1, ..., n

}and the sub-Riemannian metric is defined in such a way that this familyof vector fields is orthonormal. The symmetry V0 is given by ∂z andthe measure µ coincides with the 2n+1-dimensional Lebesgue measureL2n+1.

Corollary 2.2. The Heisenberg group satisfies

L2n+1(ϕt(U)) =

∫U

(1− t)M1(k1(x), t)M2n−22 (k2(x), t)

M1(k1(x), 0)M2n−22 (k2(x), 0)

dL2n+1(x)


k1(x) = f(x)2(V0f(x))2, k2(x) =f(x)2(V0f(x))2

4.

In particular, the metric measure space (H, d,L2n+1) satisfies the con-dition MCP(0, 0; 2n, 2n+ 1) =MCP(0, 2n+ 3).

For the complex Hopf fibration, the manifold M is the 2n+1 dimen-sional sphere M = {(z1, ..., zn+1) ∈ Cn+1||z1|2 + ...+ |zn+1|2 = 1}. Thesymmetry V0 is the infinitesimal generator of the action z 7→ e2πitz andthe distribution D is the orthogonal complement of V0 with respect tothe round metric on M .

Corollary 2.3. The complex Hopf fibration satisfies

µ(ϕt(U)) =

∫U

(1− t)M1(k1(x), t)M2n−22 (k2(x), t)

M1(k1(x), 0)M2n−22 (k2(x), 0)

dµ(x)


k1(x) = f(x)2(4 + (V0f(x))2), k2(x) =f(x)2

4

(4 + (V0f(x))2

).

In particular, the metric measure space (M,d, µ) satisfies the conditionMCP(4, 1; 2n, 2n+ 1) and hence MCP(0, 2n+ 3).


Remark 2.1. Note that if (M, 〈·, ·〉 , J) defines a Kahler manifold,

then the Tanaka-Webster connection ∇ of the corresponding Sasakianmetric on M can be defined as the following natural lift of the Levi-

Civita connection ∇ of M ([10, section 4]): Let π0 : M → M be thecanonical projection. For any horizontal (w.r.t. the distribution D)

vector fields X and Y on M , one defines ∇YX to be the unique hor-

izontal vector field on M such that (π0)∗(∇YX

)= ∇(π0)∗(Y )(π0)∗(X);

further, one sets ∇V0 = 0 and ∇V0V = [V0, V ] for any vector field Von M . This immediately implies that if RTW is the Tanaka-Webstercurvature tensor, then for any two horizontal X, Y one has

〈RTW(X, Y )Y,X〉 = 〈Rm((π0)∗(X), (π0)∗(Y )

)(π0)∗(Y ), (π0)∗X〉,

where as an inner product in the left-hand side we use the Sasakian met-ric, i.e. the extension of the sub-Riemannian metric to a Riemannianone assuming that the vector field V0 is of unit length and is orthogonalto D. Therefore Corollary 2.1 can be easily reformulated in terms ofthe Tanaka-Webster curvature tensor.

Finally, we state the corresponding Bonnet-Myer’s type Theorem inthis setting.

Theorem 2.2. Assume that the tensor J is parallel and the Riemanncurvature tensor Rm satisfies


for some positive constant k1. Then the diameter of the manifold Mwith respect to the corresponding sub-Riemannian metric is less thanor equal to 2π√

k1.

On the other hand, if λ3 + 2λ21 ≤ 0 and

Rc(v, v)− 1

|Jv|2〈Rm(Jv, v)v, Jv〉 ≥ (2n− 2)k2|v|2

for some positive constant k2, then the diameter of the manifold Mwith respect to the corresponding sub-Riemannian metric is less thanor equal to π√

k2.

The rest of the sections will be devoted to the proof of the mainresults.


3. Sub-Riemannian Geodesic Flows and MeasureContraction

In this section, we recall the definition of the sub-Riemannian ge-odesic flow and its connections with the contraction of measures ap-peared in [2, 3, 16].

As in the Riemannian case, the (constant speed) minimizers of (2.1)can be found by minimizing the following kinetic energy functional

(3.2) infγ∈Γ

∫ 1

0

1

2|γ(t)|2dt.

In the Riemannian case, the minimizers of (3.2) are given by the geo-desic equation, the Euler-Lagrange equation of the functional (3.2). Inthe sub-Riemannian case, the minimization problem in (3.2) becomes aconstrained minimization problem and it is more convenient to look atthe geodesic flow from the Hamiltonian point of view in this case. Forthis, let H : T ∗M → R be the Hamiltonian defined by the Legendretransform:

H(x,p) = supv∈D

(p(v)− 1

2|v|2).

This Hamiltonian, in turn, defines a Hamiltonian vector field ~H on thecotangent bundle T ∗M which is a sub-Riemannian analogue of the geo-desic equation. It is given, in the local coordinates (x1, ..., xn, p1, ..., pn),by

~H =n∑i=1

(Hpi∂xi −Hxi∂pi) .

We assume, through out this paper, that the vector field ~H defines

a complete flow which is denoted by et~H. In the Riemannian case,

the minimizers of (3.2) are given by the projection of the trajectories

of et~H to the manifold M . In the sub-Riemannian case, minimizers

obtained this way are called normal geodesics and they do not give allthe minimizers of (3.2) in general (see [22] for more detailed discussionson this). On the other hand, all minimizers of (3.2) are normal if thedistribution D is contact (see [22]).

Next, we discuss an analogue of the Jacobi equation in the aboveHamiltonian setting. For this, let ω be the canonical symplectic formof the cotangent bundle T ∗M . In local coordinates (x1, ..., xn, p1, ..., pn),ω is given by

ω =n∑i=1

dxi ∧ dpi.


Let π : T ∗M → M be the canonical projection and let ver thevertical sub-bundle of the cotangent bundle T ∗M defined by

ver(x,p) = {v ∈ T(x,p)T∗M |π∗(v) = 0}.

Recall that a n-dimensional subspace of a symplectic vector space isLagrangian if the symplectic form vanishes when restricted to the sub-space. Each vertical space ver(x,p) is a Lagrangian subspace of the

symplectic vector space T(x,p)T∗M . Since the flow et

~H preserves thesymplectic form ω, it also sends a Lagrangian subspace to anotherLagrangian one. Therefore, the following also forms a one-parameterfamily of Lagrangian subspaces contained in T(x,p)T

∗M

(3.3) J(x,p)(t) = e−t~H

∗ (veret~H(x,p)).

This family defines a curve in the Lagrangian Grassmannian (the spaceof Lagrangian subspaces) of T(x,p)T

∗M and it is called the Jacobi curve

at (x,p) of the flow et~H.

Assume that the distribution is contact. Then we have the followingparticular case of the results in [18, 19].

Theorem 3.3. Assume that the distribution D is contact. Then thereexists a one-parameter family of bases

E(t) = (Ea(t), Eb(t), Ec1(t), ..., Ec2n−1(t))T

F (t) = (Fa(t), Fb(t), Fc1(t), ..., Fc2n−1(t))T

of the symplectic vector space T(x,p)T∗M such that the followings hold

for any t:

(1) J(x,p)(t) = span{Ea(t), Eb(t), Ec1(t), ..., Ec2n−1(t)},(2) span{Fa(t), Fb(t), Fc1(t), ..., Fc2n−1(t)} is a family of Lagrangian

subspaces,(3) ω(Ea(t), Fa(t)) = ω(Eb(t), Fb(t)) = 1,(4) ω(Eci(t), Fcj(t)) = δij,

(5) E(t) = C1E(t) + C2F (t),(6) F (t) = −R(t)E(t)− CT

1 F (t),

where R(t) is a symmetric matrix, C1 and C2 are (2n + 1)× (2n + 1)matrices defined by

(1) C1 =

(0 10 0

)is a 2× 2 matrix,

(2) C2 =

(0 00 1

)is a 2× 2 matrix,

(3) C1 =

(C1 OO O

)


(4) C2 =

(C2 OO I

).

Moreover, a moving frame

Ea(t), Eb(t), Ec1(t), ..., Ec2n−1(t), Fa(t), Fb(t), Fc1(t), ..., Fc2n−1(t)

satisfies conditions (1)-(6) above if and only if

(3.4) (Ea(t), Eb(t), Fa(t), Fb(t)) = ±(Ea(t), Eb(t), Fa(t), Fb(t))

and there exists a constant orthogonal matrix U of size (2n−1)×(2n−1)such that for all t

(3.5) Eci(t) =2n−1∑j=1

UijEcj(t) and Fci(t) =2n−1∑j=1

UijFcj(t).

We call any frame (E(t), F (t)) in Theorem 3.3 a canonical frame atthe point (x,p) and call the equations in (5) and (6) of Theorem 3.3 thestructural equation of the Jacobi curve (3.3). Note that the conditions(3) - (4) means that the canonical frame is a family of symplectic bases.

Let us fix a point x0 in M and let f(x) = −12d2(x,x0). By the

result of [9], f is locally semi-concave in M − {x0}. In particular, it isdifferentiable almost everywhere and we can define the family of Borel

maps ϕt : M → M by ϕt(x) = π(et~H(dfx)), where 0 ≤ t ≤ 1. Note

that t 7→ ϕt(x) is a minimizing geodesic between the points x and x0

(see for instance [2]).Let M be a contact sub-Riemannian manifold with symmetry. This

means that the sub-Riemannian structure on M is defined by a con-tact distribution D and there is a sub-Riemannian isometry V0 whichis transversal to D. We can extend the sub-Riemannian on M to aRiemannian one by the conditions that the vector field V0 is of unitlength and is orthogonal to D. The corresponding Riemannian mea-sure is denoted by µ. By the result in [13], the measures (ϕt)∗µ areabsolutely continuous with respect to µ for all time t in the interval[0, 1). If (ϕt)∗µ = ρtµ, then the following equation holds on a set offull measure where f is twice differentiable:

ρt(ϕt(x)) det((dϕt)x) = 1

and the determinant is computed with respect to frames of the abovementioned Riemannian structure. Moreover, the map ϕt is invertiblefor all t in [0, 1) and so we have

(3.6) µ(ϕt(U)) =

∫U

1

ρt(ϕt(x))dµ(x) =

∫U

det((dϕt)x)dµ(x).


Therefore, in order to prove the main results and the measure contrac-tion properties, it remains to estimate det((dϕt)x) which can be doneusing the canonical frame mentioned above. The explanations on thiswill occupy the rest of this section.

Let x be a point where the function f is twice differentiable and let(E(t), F (t)) be a canonical frame at the point (x, dfx). Let

ςa = π∗(Fa(0)), ςb = π∗(Fb(0)), ςci = π∗(Fci(0)),

be the projection of the frame F (0) onto the tangent bundle TM . Letddf be the differential of the map x 7→ dfx which pushes the aboveframe on TxM to a tuple of vectors in T(x,df)T

∗M (a frame in a vectorspace is usually a basis, but the image of ς is not a basis of T(x,df)T

∗M).Therefore, we can let A(t) and B(t) be the matrices defined by

ddf(ς) = A(t)E(t) +B(t)F (t),(3.7)

where ς =(ςa, ςb, ςc1 , · · · , ςc2n−1

)Tand ddf(ς) is the column obtained by

applying ddf to each entries of ς.

Lemma 3.1. Let S(t) = B(t)−1A(t). Then

µ(ϕt(U)) ≥ 1

λ1λ2

∫U

e−∫ t0 tr(S(τ)C2)dτdµ(x),

where λ1 and λ2 are the operator norms of J and J−1, respectively.Moreover, S(t) satisfies the following matrix Riccati equation

S(t)− S(t)C2S(t) + CT1 S(t) + S(t)C1 −R(t) = 0, lim

t→1S(t)−1 = 0.

By Lemma 3.1, it remains to investigate the Riccati equation satisfiedby S(t) and the curvature matrix R(t) which will be done in next twosections.

Proof of Lemma 3.1. By (3.7) and the definition of ϕt, we have

dϕt(ς) = B(t)(π∗det~HF (t)).

Note that τ 7→ det~HF (t+ τ) is a canonical frame at et

~H(x, df). There-fore, by Lemma 6.4 below, we have

λ1|∇horf(x)|| det(dϕt)| ≥ |µ(dϕt(ς))|

= | det(B(t))µ(dπdet~HF (t))| ≥ 1

λ2

| det(B(t))||∇horf(x)|

where λ1 and λ2 are the operator norms of J and J−1, respectively. Here∇horf denotes the horizontal gradient of f defined by df(v) = 〈∇horf, v〉,where v is any vector in the distribution D.


By combining this with (3.6), we obtain

(3.8) µ(ϕt(U)) ≥ 1

λ1λ2

∫U

| det(B(t))|dµ(x).

On the other hand, by differentiating (3.7) with respect to time tand using the structural equation, we obtain

(3.9) A(t) + A(t)C1 −B(t)Rt = 0, B(t) + A(t)C2 −B(t)CT1 = 0.

Therefore,

d

dtdet(B(t)) = det(B(t))tr(B(t)−1B(t)) = − det(B(t))tr(S(t)C2).

By setting t = 0 and apply π∗ on each side of (3.7), we have B(0) = I.Therefore, we obtain

det(B(t)) = e−∫ t0 tr(S(τ)C2)dτ .

By combining this with (3.8), we obtain the first assertion.Since ϕ1(x) = x0 for all x, we have dϕ1 = 0 and so B(1) = 0. By

(3.9) and the definition of S(t), we have


t→1S(t)−1 = 0

as claimed. �

4. On the matrix Riccati equation

According to Lemma 3.1, we need to estimate the term tr(S(t)C2).In this section, we provide two such estimates which lead to the mainresults.

Throughout this section, we assume that the matrix R(t) is of theform

R(t) =

0 0 O2n−2 00 Rbb(t) Rcb(t) 0

OT2n−2 Rcb(t)

T Rcc(t) OT2n−2

0 0 O2n−2 0

,

where O2n−2 is the zero matrix of size 1 × (2n − 2). We will see thereasons for this choice in section 5.

The following is a consequence of the result in [25].

Lemma 4.2. Assume that the curvature R(t) satisfies(Rbb(t) Rcb(t)Rcb(t)

T Rcc(t)

)≥(

k1 00 k2I

),


where I is of size (2n−2)×(2n−2), k1 and k2 are two constants. Then

e−∫ t0 tr(C2S(τ))dτ ≥ (1− t)2n+3M1(k1, t)M2n−2

2 (k2, t)

M1(k1, 0)M2n−22 (k2, 0)

,

where

D(k, t) =√|k|(1− t),

M1(k, t) =

2−2 cos(D(k,t))−D(k,t) sin(D(k,t))

D(k,t)4if k > 0

112

if k = 02−2 cosh(D(k,t))+D(k,t) sinh(D(k,t))

D(k,t)4if k < 0,

M2(k, t) =

sin(D(k,t))D(k,t)

if k > 0

1 if k = 0sinh(D(k,t))D(k,t)

if k < 0.

Moreover, equality holds if(Rbb(t) Rcb(t)Rcb(t)

T Rcc(t)

)=

(k1 00 k2I

),

Proof. We only prove the case when both constants k1 and k2 are posi-tive. The proofs for other cases are similar and are therefore omitted.Recall that S(t) satisfies


t→1S(t)−1 = 0.

Let Γ(t) be a solution of the following

Γ(t)− Γ(t)C2Γ(t) + CT1 Γ(t) + Γ(t)C1 −K = 0, lim

t→1Γ(t)−1 = 0,

where K =

(k1 00 k2I

).

By the result of [25], S(t) ≤ Γ(t). Therefore, tr(C2S(t)) ≤ tr(C2Γ(t)).On the other hand, Γ(t) can be computed using the result in [16] andit follows that

tr(C2S(t)) ≤√k1(sin(D(k1, t))−D(k1, t) cos(D(k1, t)))

2− 2 cos(D(k1, t))−D(k1, t) sin(D(k1, t))

+ (2n− 2)√

k2 cot(D(k2, t)) +1

1− t.

The assertion follows from integrating the above inequality. �

Next, we consider the case where the assumptions are weaker thanthose in Lemma 4.2.


Lemma 4.3. Assume that the curvature R(t) satisfies Rbb(t) ≥ k1 andtr(Rcc(t)) ≥ k2(2n− 2) for some constants k1 and k2. Then

e−∫ t0 tr(C2S(τ))dτ ≥ (1− t)2n+3M1(k1, t)M2n−2

2 (k2, t)

M1(k1, 0)M2n−22 (k2, 0)

.

Proof. Once again, we only prove the case when both constants k1 andk2 are positive. Let us write

S(t) =

S1(t) S2(t) S3(t)S2(t)T S4(t) S5(t)S3(t)T S5(t)T S6(t)

,

where S1(t) is a 2× 2 matrix and S6(t) is 1× 1. Then

S1(t)− S1(t)C2S1(t)− S2(t)S2(t)T

− S3(t)2 + CT1 S1(t) + S1(t)C1 −R1(t) = 0,

S4(t)− S4(t)2 − S5(t)S5(t)T − S2(t)T C2S2(t)−Rcc(t) = 0,

S6(t)− S6(t)2 − S5(t)TS5(t)− S3(t)T C2S3(t) = 0,

(4.10)

where C1 =

(0 10 0

), C2 =

(0 00 1

), and R1(t) =

(0 00 Rbb(t)

).

By the same argument as in Lemma 4.2, we have

tr(C2S1(t)) ≤√k1(sin(D(k1, t))−D(k1, t) cos(D(k1, t)))

2− 2 cos(D(k1, t))−D(k1, t) sin(D(k1, t)).(4.11)

For the term S4(t), we can take the trace and obtain

d

dttr(S4(t)) ≥ 1

2n− 2tr(S4(t))2 + (2n− 2)k2.

Therefore, an argument as in Lemma 4.2 again gives

(4.12) trS4(t) ≤√|k2|(2n− 2) cot (D(k2, t)) .

Finally, for the term S6(t), we also have

S6(t) ≥ S6(t)2.

Therefore,

S6(t) ≤ 1

1− t.

By combining this with (4.11) and (4.12), we obtain

tr(C2S(t)) ≤√|k2|(2n− 2) cot (D(k2, t)) +

1

1− t

+

√k1(sin(D(k1, t))−D(k1, t) cos(D(k1, t)))

2− 2 cos(D(k1, t))−D(k1, t) sin(D(k1, t)).


The rest follows as in Lemma 4.2. �

5. Curvature of Contact Sub-Riemannian manifolds withsymmetry

Let (E(t), F (t)) be a canonical frame at a point (x,p) of the cotan-gent bundle T ∗M . Recall that the vertical bundle ver of TT ∗M isgiven by

ver = {V ∈ TT ∗M |π∗V = 0}.The linear map R : ver → ver having the matrix R(0) with respectto the canonical frame (E(0), F (0)) is called the curvature map. Moreprecisely,

R(x,p)(V ) = AR(0)E(0),

where V = AE(0) and A is any row vectors of suitable size. Moreoverit follows from Theorem 3.3 that the above definition does not dependon the choice of the canonical frame. In this section, we discuss R inthe case where the manifold M is a contact sub-Riemannian manifoldwith symmetry.

Let α be a contact form of the given distribution D. Then therestriction of dα onto the distribution D is a non-degenerate 2-form.This defines a skew-symmetric linear bundle map J : D → D by

dα(X, Y ) = 〈JX, Y 〉for any pair of vectors X and Y contained in D. In addition assumethat the Reeb field V0 is an infinitesimal symmetry, i.e.

etV0∗ D = D , (etV0)∗ 〈·, ·〉 = 〈·, ·〉 .Assume also that V0 is transversal to the distribution D.

We also assume that the quotient M of the manifold M by the sym-

metry V0 is also a manifold. The quotient map π0 : M → M defines

an identification of D with TM . Therefore, both the sub-Riemannian

metric 〈·, ·〉 and the map J descend to TM which are denoted by thesame symbol. We also let u0 : T ∗M → R be the function defined byu0(x,p) = p(V0(x)).

Next, we introduce some notations for later use. We recall the iden-tification of the cotangent space T ∗xM and the vertical space ver(x,p)

by

q 7→ qver :=d

dt(p + tq)

∣∣∣t=0,

where p and q are covectors in T ∗xM . By using the above identification,we can assign a unique covector q in T ∗xM to each vector v in thevertical bundle ver(x,p) such that qver = v. This, in turn, defines


a vector v in the distribution D by q(w) = 〈v, w〉, where w is any

vector in the distribution D. Finally, the vector vh in TM is definedby vh = (π0)∗v. We will also denote this vector by qh.

The linear map q 7→ I(q) := qh = vh is surjective with a 1-

dimensional kernel. Therefore, given any vector X in Tπ0(x)M , there isa 1-dimensional affine subspace of the cotangent space T ∗xM such thatany covector q inside satisfies I(q) = vh. Moreover, there is a uniquecovector q0 in this affine space which satisfies the condition q0(V0) = 0.Here V0 is the symmetry introduced earlier. Finally, we denote by Xv

the vector qver0 in the vertical space T(x,p)T

∗M .The frame E(0) = (Ea(0), Eb(0), Ec1(0), ..., Ec2n−1(0))T defines a split-

ting of the vertical space ver = vera ⊕ verb ⊕ verc, which are charac-terized as follows (see [19, Section 3] for a proof). Let ∂u0 be the vectorfield in vera satisfying the condition du0(∂u0) = 1.

Proposition 5.1. The subspaces vera,verb and verc are given by thefollowings:

(1) vera := span{Ea(0)} = span{Ea :=∂u0|Jph|},

(2) verb := span{Eb(0)} = span{Eb := (Jph)v

|Jph| + ~H(

1|Jph|

)∂u0

},

(3) verc := span{Ec1(0), ..., Ec2n−1(0)}= {Xv +A(Xv)

∂u0|Jph| |X ∈ span{Jph}⊥},

where A is a linear functional on verc defined by A(v) = −⟨vh,Vh

1

⟩and V1 ∈ verc such that

Vh1 = − 2

|Jph|∇phJ(ph) +

u0

|Jph|J2ph

+u0|Jph||ph|2

ph +2

|Jph|3⟨∇phJ(ph), Jph

⟩Jph.

Let V be a vector in verz1 , where z1 = a, b, c1, ..., c2n−1. The verz2-component of R(V ) is denoted by R(z1, z2)(V ). In other words,

R(V ) = R(z1, a)(V ) + R(z1, b)(V ) + R(z1, c)(V ),

where R(z1, a)(V ), R(z1, b)(V ), and R(z1, c)(V ) are contained in vera,verb, and verc, respectively. The following theorems follows from [19]

(see Appendix I for details).

Theorem 5.4. The components R(c, c) and R(b, c) of the curvatureR satisfy the followings:

(1)⟨(

R(c, c)(v))h,vh⟩

=⟨Rm(vh,ph)ph,vh

⟩+ u0(x,p)

⟨vh,∇vhJ(ph)

⟩+

u20(x,p)

4|Jvh|2 − 1

4A(v)2 ,


(2) R(c, b)v = ρ(c, b)(v)Eb,(3) ρ(c, b)(v) = 1

|Jph|

⟨Rm(Jph,ph)ph,vh

⟩− 3|Jph|

⟨vh,∇2J(ph,ph,ph)

⟩+ 4u0(x,p)

|Jph|

⟨vh,∇phJ(Jph) +∇JphJ(ph)

⟩+

u20(x,p)

|Jph|

⟨Jvh, J2ph

⟩+ 8|Jph|3

⟨Jph,∇phJ(ph)

⟩ ⟨vh,∇phJ(ph)

⟩− 4u0(x,p)

|Jph|3⟨Jph,∇phJ(ph)

⟩ ⟨vh, J2ph

⟩.

Remark 5.2. In Theorem 5.4, we use the following definition of theRiemann curvature tensor

Rm(X, Y )Z = ∇X∇YZ −∇Y∇XZ −∇[X,Y ]Z.

This is different from that of [19] by a minus sign.

Theorem 5.5. The component R(b, b) of the curvature R satisfies thefollowings:

(1) R(b, b)Eb = ρ(b, b)Eb,(2) ρ(b, b) = 1

|Jph|2⟨Rm(ph, Jph)Jph,ph

⟩− 10|Jph|4

⟨∇phJ(ph), Jph

⟩2

+ 6|Jph|2 |∇phJ(ph)|2 + 3

|Jph|2⟨Jph,∇2J(ph,ph,ph)

⟩− 2u0(x,p)

|Jph|2⟨Jph,∇JphJ(ph)

⟩− 3u0(x,p)

|Jph|2⟨Jph,∇phJ(Jph)

⟩− 6u0(x,p)

|Jph|2⟨J2ph,∇phJ(ph)

⟩+

u20(x,p)

|Jph|2 |J2ph|2,

Let Kt be the linear maps from T(x,p)(T∗M) to Tet~H(x,p)T

∗M , sending

E(0) to(et~H)∗E(t) and F (x,p)(0) to

(et~H)∗F (t). The map Kt is called

the parallel transport along the curve et~H(x,p) at time t. Note that Kt

sends the vertical space ver(x,p) to the vertical space veret~H(x,p).

Let S : ver→ R be a function on the vertical bundle ver. Then theith derivative of S along a path t 7→ Kt(v) is denoted by S(i)(v). Moreprecisely, we have

S(i)(v) =di

dtiS(Kt(v))

∣∣∣t=0.

In the following theorem, we need this notation for the function v 7→A(v). The explicit expressions of the derived maps A(1),A(2) (in termsof the tensor J and its covariant derivatives) are given by Proposition9.3 in Appendix II.

Theorem 5.6. The curvature maps R(c, a) and R(a, a) are given by

(1) R(c, a)v = ρ(c, a)(v)∂u0|Jph| ,

(2) ρ(c, a)v = A(2)(v) + 2|Jph|~H(

1|Jph|

)A(1)(v)

+|Jph|~H2(

1|Jph|

)A(v)−

⟨(R(c, c)v

)h,Vh

1

⟩+|Jph|~H

(1|Jph|

)ρ(c, b)v,


(3) R(a, a)∂u0 = ρ(a, a)∂u0,

(4) ρ(a, a) = ~H (ρ(c, b)(V1)) + |Jph|~H(

1|Jph|

)~H(ρ(b, b))

+ ρ(c, a)(V1)− |Jph|~H(

1|Jph|

)ρ(c, b)(V1)

+ |Jph|~H2(

1|Jph|

)ρ(b, b) + |Jph|~H4

(1|Jph|

).

Here V1 and Vh1 are as in Proposition 5.1.

The curvature R is much simpler when J is parallel, i.e. ∇J = 0.

Corollary 5.4. Assume that ∇J = 0. Then

(1)⟨(

R(c, c)(v))h,vh⟩

=⟨Rm(vh,ph)ph,vh

⟩+

u20(x,p)

4

(|Jvh|2 − 1

|Jph|2⟨vh, J2ph

⟩2)

,

(2) ρ(c, b)v = 1|Jph|

⟨Rm(Jph,ph)ph,vh

⟩+

u20(x,p)

|Jph|

⟨Jvh, J2ph

⟩,

(3) ρ(b, b) = 1|Jph|2

⟨Rm(Jph,ph)ph, Jph

⟩+

u20(x,p)

|Jph|2 |J2ph|2,

(4) R(c, a) = 0,(5) R(a, a) = 0.

If, in addition to the condition ∇J = 0, we assume that J2 = −I(i.e. the manifold M equipped with (〈·, ·〉 , J) is a Kahler manifold),then

Corollary 5.5. Assume that ∇J = 0 and J2 = −I. Then

(1)⟨(R(c, c)(v))h,vh

⟩=⟨Rm(vh,ph)ph,vh

⟩+

u20(x,p)

4|vh|2,

(2) ρ(c, b)(v) = 1|ph|

⟨Rm(Jph,ph)ph,vh

⟩,

(3) ρ(b, b) = 1|ph|2

⟨Rm(Jph,ph)ph, Jph

⟩+ u2

0(x,p),

(4) R(c, a) = 0,(5) R(a, a) = 0.

In general, the expression for ρ(a, a) is very complicated and it isdifficult to check whether it is nonnegative or not. Instead, we onlymention the following partial result (see Appendix II for the proof).

Theorem 5.7. The invariant ρ(a, a) is a cubic polynomial on u0. Inaddition, if we assume that J2 = −I (i.e. (〈·, ·〉 , J) is an almost Kahler

structure on M), then the coefficient of the u30-term in ρ(a, a) vanishes

and the u20-term is

4u20(x,p)

|ph|2|∇phJ(Jph)|2 +

8u20(x,p)

|ph|2|∇JphJ(ph)|2

+12u2

0(x,p)

|ph|2⟨∇phJ(Jph),∇JphJ(ph)

⟩.


Remark 5.3. Theorem 5.7 shows that one can use the method in thispaper to study measure contraction property for almost Kahler mani-folds.

Now let us investigate the component R(c, c) in order to apply Lemma4.3. As a direct consequence of Theorem 5.4, we obtain

Theorem 5.8. Let Rc be the Ricci curvature tensor of (M, 〈·, ·〉).Then

tr(R(c, c)) = Rc(ph,ph)− 1

|Jph|2⟨Rm(Jph,ph)ph, Jph

⟩+ tr(T(x,p)),

where T(x,p) is a (1, 1)-tensor on M defined by

T(x,p) = − 1

|Jph|2⟨vh,∇phJ(ph)

⟩∇phJ(ph)

+ u0(x,p)∇vhJ(ph) +u0(x,p)

|Jph|2⟨vh,∇phJ(ph)

⟩J2ph

− 1

4u2

0(x,p)J2vh − u20(x,p)

4|Jph|2⟨vh, J2ph

⟩J2ph.

Corollary 5.6. In the almost Kahlerian case (J2 = −I),


|ph|2⟨Rm(Jph,ph)ph, Jph

⟩+n− 1

2u2

0(x,p)−|∇phJ(ph)|2

|ph|2+ u0(x,p)tr

⟨∇vhJ(ph),vh

⟩,

where tr is the trace taken with respect to the Riemannian metric on

the quotient M .

Corollary 5.7. Assume that J is parallel. Then


|Jph|2⟨Rm(Jph,ph)ph, Jph

⟩− u2

0(x,p)

4

(tr(J2) + 2

|J2ph|2

|Jph|2

).

Corollary 5.8. Assume that J2 = −I and ∇J = 0 (i.e. the Kahlercase),


|ph|2⟨Rm(Jph,ph)ph, Jph

⟩+n− 1

2u2

0(x,p).


6. Proof of the Main Results

In this section, we finish the proof of the main results. Let E(t), F (t)be a canonical frame at a point (x,p) in the cotangent bundle T ∗M .Let Ez and Fz be two vector fields defined locally near (x,p) and satisfythe following conditions

(6.13) Ez(et~H(x,p)) = et

~H∗ Ez(t), Fz(et

~H(x,p)) = et~H∗ Fz(t),

where z = a, b, c1, ..., c2n−1.We also let

ςz = π∗(Fz(x,p)).

We start with the following lemma which was needed in the proof ofLemma 3.1.

Lemma 6.4. The above frame satisfies the followings:

(1) ςb, ςc1 , ςc2 ..., ςc2n−1 is orthonormal with respect to the sub-Riemannianmetric 〈·, ·〉,

(2) µ(ςa, ςb, ςc1 , ..., ςc2n−1) = |Jph|.

Proof. A computation shows that for any vector field V contained inthe vertical bundle ver, there holds

(π0 ◦ π)∗([~H, V ]) = −V h,

where we recall that π0 : M → M is the quotient map. On the otherhand, from Theorem 3.3 and (6.13), we have

Fb(x,p) =d

dte−t

~H∗ Eb(et

~H(x,p))∣∣∣t=0

= [~H, Eb](x,p),

and

Fci(x,p) =d

dte−t

~H∗ Eci(et

~h(x,p))∣∣∣t=0

= [~H, Eci ](x,p),

where i = 1, ..., 2n− 1.It follows that

(π0)∗(ςb(x,p)) = −(Eb(x,p))h

and(π0)∗(ςci(x,p)) = −(Eci(x,p))h,

where i = 1, ..., 2n− 1.

Since (π0)∗ : D → TM is a Riemannian isometry, it suffices to show

(Eb(x,p))h, (Ec1(x,p))h, ..., (Ec2n−1(x,p))h

is orthonormal. Indeed, it follows from Theorem 3.3 that

〈(Eci(x,p))h, (Ecj(x,p))h〉 = ω(Eci(x,p),Fcj(x,p))


= ω(Eci(0), Fcj(0)) = δij.

The orthonormal relations involving (Eb(x,p))h are proved in a similarway. This finishes the proof of the first assertion.

For the second assertion, it suffices to show that ςa = −|Jph|V0.From Theorem 3.3, it follows

〈(Eb(x,p))h, (π0)∗ςa〉 = −ω(Eb(x,p),Fa(x,p)) = 0,

〈(Eci(x,p))h, (π0)∗ςa〉 = −ω(Eci(x,p),Fa(x,p)) = 0.

Hence (π0)∗ςa = 0 and so we can assume ςa is contained in the subspacespanned by V0.

Let ~u0 be the Hamiltonian vector field of the Hamiltonian u0. Since(π0)∗~u0 = V0, we can assume

Fa(x,p) = f~u0 + V,

where f is a function in the cotangent bundle T ∗M and V is a vectorin the vertical bundle.

By combining Theorem 3.3 and Proposition 5.1, we have

1 = ω(Ea(0), Fa(0)) = ω(Ea(x,p),Fa(x,p))

= ω

(∂u0|Jph|

, f~u0

)= − f

|Jph|.

Hence f = −|Jph| as claimed. �

Next, we give the proof of Theorem 2.1. Note that Corollary 2.1 is animmediate consequence of Theorem 2.1 and Corollary 5.8. Corollary2.2 and 2.3 are also consequences of the proof of Theorem 2.1 andequality case of Lemma 4.2.

Proof of Theorem 2.1. If E(t), F (t) is a canonical frame at the point(x, dfx) in the cotangent bundle T ∗M , then

t 7→ (deτ~H(E(t+ τ)), deτ

~H(F (t+ τ)))

is a canonical frame at the point et~H(x, dfx). It follows from this that

R(t) is the matrix representation of the operator Ret~H(x,df) with respect

to the frame deτ~H(E(τ)).

Since V0 is a symmetry, u0 is constant along the flow et~H (see for

instance [22]). Therefore, by the assumptions and Lemma 5.7, Rbb(t)

and tr(Rcc(t)) are bounded below by kb|∇horf|2 +u20(x,df)

λ22and kc(2n−

2)|∇horf|2−u20(x,df)

4(λ3 + 2λ2

1), respectively. Therefore, the assumptionsof Lemma 4.3 are satisfied. By combining this with Lemma 3.1, theresult follows. �


7. Proof of Bonnet-Myer’s type Theorem

In this section, we prove Theorem 2.2.

Proof of Theorem 2.2. Let (x,p) be the covector in T ∗M such that

the corresponding sub-Riemannian geodesic γ(t) = π(et~H(x,p)) with

0 ≤ t ≤ 1 is minimizing and the length is equal to the diameter of Mwith respect to the sub-Riemannian metric. It follows that (see [4])

det~H(ver(x,p)) intersects veret~H(x,p) transversely for all t in (0, 1). Let

E(t), F (t) be a canonical frame at the point (x,p). Let A(t) and B(t)be matrices defined by

E(0) = A(t)E(t) +B(t)F (t).

It follows from the above discussion that B(t) is invertible for all t in(0, 1). Clearly, we also have A(0) = I and B(0) = 0. Therefore, ifS(t) = B(t)−1A(t), then we have

S(t)− S(t)C2S(t) + CT1 S(t) + S(t)C1 −R(t) = 0

with limt→0 S(t)−1 = 0.Assume that k1 > 0. Then, by arguing as in Lemma 4.3, we have

tr(C2S1(t)) ≥ −√k1(sin(t

√k1)− t

√k1 cos(t

√k1))

2− 2 cos(t√k1)− t

√k1 sin(t

√k1)

(7.14)

where

f(y) = d(x, y),

k1(y) = f(y)2

(k1 +

(V0f)2(y)

λ22

)≥ k1f(y)2,

S(t) =

S1(t) S2(t) S3(t)S2(t)T S4(t) S5(t)S3(t)T S5(t)T S6(t)

,

C2 =

(0 00 1

).

Similarly, if k2 > 0, then

(7.15) trS4(t) ≥ −√

k2(2n− 2) cot(t√

k2

),

where

k2(y) = f(y)2

(k2 −

(V0f)2(y)

4(2n− 2)

(λ3 + 2λ2

1

))≥ k2f(y)2.


Therefore, if 2π√k1d(x,y)

< 1 or π√k2d(x,y)

< 1, we obtain a contradiction

since the right hand sides of (7.14) and (7.15) go to ∞ for some timet in (0, 1) in this case. �

8. Appendix I: Proofs of Theorems 5.4-5.6

First, we introduce another version of Jacobi curves J(·) and thecurvature R, called reduced Jacobi curves and reduced curvature, re-spectively. Then we show that the curvature R can be recovered fromthe curvature R (see Theorem 8.10 below), which make Theorems 5.4-5.6 to be the consequences of the results on the curvature R in [19].

Recall that the Hamiltonian H is constant along the flow et~H, so we

can define another curve J(x,p), called reduced Jacobi curve at (x,p),by

(8.16) t 7−→ J(x,p)(t) :=(J(x,p)(t) ∩ ~H∠(x,p)

)/R~H(x,p).

Here ~H∠ denotes the skew orthogonal complement of ~H with respectto the symplectic form ω.

The symplectic form ω descends to a symplectic form on ~H∠(x,p)/R~H(x,p)

and the reduced Jacobi curve J(x,p) is a curve in the Lagrange Grass-

mannian of ~H∠(x,p)/R~H(x,p). Conversely, we can recover the non-reducedJacobi curve from the reduced one. Indeed, the vertical space ver(x,p)

splits into the following direct sum

ver(x,p) =(ver(x,p) ∩ ~H∠(x,p)

)⊕ RE(x,p).

Here E is the Euler field defined in local coordinates by E =∑

i pi∂pi .

A computation shows that et~H∗ E = E− t~H. It follows that

(8.17) J(x,p)(t) =(J(x,p)(t) ∩ ~H∠(x,p)

)⊕ R

(E− t~H

).

From the right hand side of (8.17), it is clear that the curve J(x,p) iscompletely determined by the reduced Jacobi curve J(x,p) as claimed.

In a similar way, we have the following descriptions of a normalmoving frames {E(t), F (t)} of the reduced Jacobi cuves Jx,p also as aparticular case of the results in [17, 18].

Theorem 8.9. Assume that the distribution D is contact. Then thereexists a one-parameter family of bases

E(t) = (Ea(t), Eb(t), Ec1(t), ..., Ec2n−2(t))T

F (t) = (Fa(t), Fb(t), Fc1(t), ..., Fc2n−2(t))T


of the symplectic vector space

~H∠(x,p)/R~H(x,p)

such that the followings hold for any t:

(1) J(x,p)(t) = span{Ea(t), Eb(t), Ec1(t), ..., Ec2n−2(t)},(2) span{Fa(t), Fb(t), Fc1(t), ..., Fc2n−2(t)} is a family of Lagrangian

subspaces,(3) ω(Ea(t), Fa(t)) = ω(Eb(t), Fb(t)) = 1,(4) ω(Eci(t), Fcj(t)) = δij,

(5) ˙E(t) = C1E(t) + C2F (t),

(6) ˙F (t) = −R(t)E(t)− CT1 F (t),

where R(t) is a symmetric matrix, C1 and C2 are 2n × 2n matricesdefined by

(1) C1 =

(0 10 0

)is a 2× 2 matrix,

(2) C2 =

(0 00 1

)is a 2× 2 matrix,

(3) C1 =

(C1 OO O

)(4) C2 =

(C2 OO I

).

Moreover, a moving frame

Ea(t), Eb(t), Ec1(t), ..., Ec2n−1(t), Fa(t), Fb(t), Fc1(t), ..., Fc2n−1(t)

satisfies conditions (1)-(5) above if and only if

(8.18) (Ea(t), Eb(t), Fa(t), Fb(t)) = ±(Ea(t), Eb(t), Fa(t), Fb(t))

and there exists a constant orthogonal matrix U of size (2n−2)×(2n−2)such that for all t

(8.19) Eci(t) =2n−2∑j=1

UijEcj(t) and Fci(t) =2n−2∑j=1

UijFcj(t).

Similar to the case of a non-reduced Jacobi curve, the linear map

R : (ver(x,p) ∩ ~H∠(x,p))/R~H(x,p)→ (ver(x,p) ∩ ~H∠(x,p))/R~H(x,p)

having the matrix R(0) with respect to the canonical frame (E(0), F (0))is called the reduced curvature map.

Next, we investigate how the non-reduced curvature can be recoveredfrom the reduced curvature. Since ~H is transversal to the verticalspaces, we can identify J(x,p)(0) with its representative in J(x,p)(0) ∩


~H∠(x,p). Therefore, we can consider R as a linear map on ver(x,p) ∩~H∠(x,p).

Theorem 8.10. The curvature R can be recovered from the operatorR as follows:

(1) R|ver(x,p)∩~H∠(x,p)

= R;

(2) R(E) = 0.

Proof. Let {E(t), F (t)} be a normal moving frame (in ~H∠(x,p)/R~H(x,p))

for the curve J. Under the aforementioned identification between thespaces J(x,p) and J(x,p) ∩ ~H∠(x,p), E(t) corresponds to a basis of J(x,p) ∩~H∠(x,p), which will be denoted by the same symbol.

As before, write

E(t) = (Ea(t), Eb(t), Ec1(t), ..., Ec2n−2(t))T ,

Ec(t) = (Ec1(t), ..., Ec2n−2(t)).(8.20)

Then set

Fb(t) := ˙Eb(t)

Fc(t) := ˙Ec(t)

Fa(t) := − ˙Fb(t)− Eb(t)Rt(b, b)− Ec(t)Rt(b, c),

where Rt(·, ·) are as in Theorem 8.9.

Further, let ε(t) = (e−t~H)∗E(et

~H(x,p)), then, by a computation, we

have υ0(t) := ε(t) = ~H(x,p) and so υ0(t) = 0. Next, we let Ec(t) :=

{Ec(t), ε0(t)}, Fc(t) = {Fc(t), υ0(t)}. We will show that the tuple

(8.21) {Ea(t), Eb(t), Ec(t), Fa(t), Fb(t), Fc(t)}.

constitute a normal moving frame for non-reduced Jacobi curve J(x,p).Let us first show that the frame (8.21) is a symplectic moving frame.

By construction, it is sufficient to check that ε0(t) is skew orthogonal tothe vectors Fa(t), Fb(t), and the vectors from the tuple Fc(t). Indeed,since J(x,p) is Lagrangian, one has ω(ε0(t), Eb(t)

)= 0. By differentiating

this identity , we get

(8.22) ω(ε0(t), Eb(t))+ω(ε0(t), Fb(t)

)= 0.

Since ε0(t) = ~H(x,p) and Eb(t) ∈ ~H(x,p)∠, the first term of (8.22) isequal to zero, which gives

(8.23) ω(ε0(t), Fb(t))= 0.


The identity ω(ε0(t), Fc(t)

)= 0 is proved in completely the same

way. For the proof of ω(ε0(t), Fa(t)

)= 0, it is enough to prove that

ω(ε0(t), ˙Fb(t)

)= 0. The latter follows by the same scheme by differen-

tiating (8.23).Furthermore, the moving symplectic frame satisfies the following

structure equations

(8.24)

˙Ea(t) = Eb(t)˙Eb(t) = Fb(t)˙Ec(t) = Fc(t)

ε0(t) = υ0(t)˙Fa(t) = −Ea(t)Rt(a, a)− Ec(t)Rt(a, c)˙Fb(t) = −Eb(t)Rt(b, b)− Ec(t)Rt(b, c)− Fa(t)˙Fc(t) = −Ea(t)Rt(c, a)− Eb(t)Rt(c, b)− Ec(t)Rt(c, c)

υ0(t) = 0

This yields that the moving frame (8.21) is normal for the curve J(x,p)

(see Theorem 3.3). This and the form of the structure equation (8.24)implies the statement of of the present theorem due to the uniquenesspart of Theorem 3.3. �

Now the results in Theorems 5.4-5.6 are consequences of the calcu-lations on the reduced curvature R in Section 5 of [19].

9. Appendix II: Proof of Theorem 5.7

Since the tensor J is defined using a closed 2-form dα, we have

Proposition 9.2. For any vector fields X, Y, Z on M ,

(1) 〈X,∇ZJ(Y )〉+ 〈Y,∇XJ(Z)〉+ 〈Z,∇Y J(X)〉 = 0,(2) 〈X,∇ZJ(Y )〉+ 〈Y,∇ZJ(X)〉 = 0.

The following two propositions are consequences of definition of A(c.f. Theorem 5.4) and Proposition 4.2 in [19] .

Proposition 9.3. Let v be a vector in verc. Then

A(1)(v) = −A(v)A(

(Jph)v

|Jph|

)+

1

|Jph|⟨vh, 2∇2J(ph,ph,ph)

⟩+

1

|Jph|

⟨vh,−3u0∇phJ(Jph)− 2u0∇JphJ(ph) +

1

2u2

0J3ph⟩,


Proposition 9.4. Let v be a vector in verc. Then

A(2)(v) = −A(1)(v)A(

(Jph)v

|Jph|

)−A(v)A(1)

((Jph)v

|Jph|

)+ ~H

(1

|Jph|

)⟨vh, 2∇2J(ph,ph,ph)

⟩− 1

|Jph|2⟨Jph,∇2J(ph,ph,ph)

⟩A(v)

+1

|Jph|⟨vh, 2∇3J(ph,ph,ph,ph)

⟩− u0

|Jph|⟨vh, 2∇2J(Jph,ph,ph) + 2∇2J(ph, Jph,ph)

+2∇2J(ph,ph, Jph)− J∇2J(ph,ph,ph)⟩

+ ~H

(1

|Jph|

)⟨vh,−3u0∇phJ(Jph)− 2u0∇JphJ(ph) +

1

2u2

0J3ph⟩

− A(v)

2|Jph|2

⟨Jph,−3u0∇phJ(Jph)− 2u0∇JphJ(ph) +

1

2u2

0J3ph⟩

+1

|Jph|⟨vh,−3u0∇2J(Jph,ph,ph)− 3u0∇phJ(∇phJ(ph))

⟩+

1

|Jph|

⟨vh,−2u0∇2J(ph, Jph,ph)− 2u0∇∇

phJ(ph)J(ph)⟩

+1

2|Jph|⟨vh, u2

0∇ph(J2ph) + u20J∇ph(Jph) + u2

0J2∇ph(ph)

⟩+

1

|Jph|⟨vh, 3u2

0∇JphJ(Jph) + 3u20∇phJ(J2ph) + 2u2

0∇J2phJ(ph)

+2u20∇JphJ(Jph)− 1

2u3

0J4ph⟩

+1

2|Jph|

⟨vh,−3u2

0J∇phJ(Jph)− 2u0J∇JphJ(ph) +1

2u3

0J4ph⟩

From now on, we assume J2 = −I. Then in addition to Proposition9.2, we also have

Proposition 9.5. For any vector fields X, Y on M ,

(1) ∇Y J(JX) + J∇Y J(X) = 0,

(2) 〈JX,∇XJ(X)〉 = 0.


Proof. The first item is from taking covariant derivative of J2. For thesecond item, as Riemannian metric is compatible, then

〈JX,∇XJ(X)〉 = 〈JX,∇XJX − J∇XX〉= 〈JX,∇XJX〉 − 〈X,∇XX〉

=1

2X(|JX|2)− 1

2X(|X|2) = 0.

�

Next, we show that the u30−term in ρ(a, a) vanishes. From Proposi-

tion 9.3 and Proposition 9.5 it follows that⟨J2ph,∇phJ(Jph) +∇JphJ(ph) + J∇Jph(ph)

⟩= 0.

and ⟨J2ph,∇J2phJ(ph)

⟩+⟨J2ph,∇phJ(J2ph)

⟩+⟨J4ph,∇phJ(ph)

⟩= 0

With the above two identities, together again that J2 = −I, the claimfollows.

Finally, we analyze the u20-term of ρ(a, a). Note that in the present

case, the vector V1 contained in verc satisfies

Vh1 = − 2

|ph|∇phJ(ph).

Since |Jph| = |ph| =√

2H,

(9.25) ρ(a, a) = ~H (ρ(c, b)(V1)) + ρ(c,a)(V1).

From [19, Proposition 4.2], Propositions 9.3, and 9.5, one can get (aftersome calculations) that the u2

0-term in ρ(c, a)v is

6u20

⟨vh,−∇phJ(ph) +∇JphJ(Jph)

⟩.

Furthermore, the only u0-term of ρ(c,b)(∇phJ(ph)) is

4u0

⟨∇phJ(ph),∇phJ(Jph) +∇JphJ(ph)

⟩.

By combining the above analysis with identity (9.25), we get the con-clusion on u0-term in ρ(a, a) and thus complete the proof of Theorem5.7.


References

[1] A.A. Agrachev, R.V. Gamkrelidze: Feedback-invariant optimal control theory- I. Regular extremals, J. Dynamical and Control Systems, 3, No. 3, 343-389(1997).

[2] A. Agrachev, P. Lee: Generalized Ricci curvature bounds for three dimen-sional contact subriemannian manifolds, arXiv: 0903.2550 (2009), 31pp.

[3] A. Agrachev, P. Lee: Bishop and Laplacian comparison theorems on three di-mensional contact subriemannian manifolds with symmetry, arXiv: 1105.2206(2011), 25pp.

[4] A. Agrachev, Y. Sachkov: Control theory from the geometric viewpoint. En-cyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization,II. Springer-Verlag, Berlin, 2004.

[5] A. Agrachev, I. Zelenko: Geometry of Jacobi curves. I, J. Dynamical andControl systems, 8,No. 1, 93-140 (2002).

[6] E. Barletta, S. Dragomir: Jacobi fields of the Tanaka-Webster connection onSasakian manifolds, Kodai Math. J., 29, Number 3 (2006), 406-454.

[7] F. Baudoin, N. Garofalo: Generalized Bochner formulas and Ricci lowerbounds for sub-Riemannian manifolds of rank two, preprint, arXiv:0904.1623.

[8] F. Baudoin, N. Garofalo: Curvature-dimension inequalities and Ricci lowerbounds for sub-Riemannian manifolds with transverse symmetries, preprint,arXiv:1101.3590.

[9] P. Cannarsa, L. Rifford: Semiconcavity results for optimal control problemsadmitting no singular minimizing controls. Ann. Inst. H. Poincare Anal. NonLineaire 25 (2008), no. 4, 773–802.

[10] B. Cappelletti Montano: Some remarks on the generalized Tanaka-Websterconnection of a contact metric manifold. Rocky Mountain J. Math. 40 (2010),no. 3, 1009–1037.

[11] S. Chanillo, P. Yang: Isoperimetric inequalities & volume comparison theo-rems on CR manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no.2, 279-307.

[12] T. Coulhon, I. Holopainen, L. Saloff-Coste: Harnack inequality and hyperbol-icity for subelliptic p-Laplacians with applications to Picard type theorems.Geom. Funct. Anal. 11 (2001), no. 6, 1139-1191.

[13] A. Figalli, L. Rifford: Mass Transportation on sub-Riemannian Manifolds.Geom. Funct. Anal. 20 (2010), no. 1, 124?-159.

[14] N. Juillet: Geometric Inequalities and Generalized Ricci Bounds in theHeisenberg Group, Int. Math. Res. Not. IMRN 2009, no. 13, 2347?-2373.

[15] P.W.Y. Lee: Displacement interpolations from a Hamiltonian point of view,arXiv:1205.1442 (2012), 34pp.

[16] J.J. Levin: On the matrix Riccati equation. Proc. Amer. Math. Soc. 10 (1959)519–524.

[17] C. Li, I. Zelenko: Parametrized curves in Lagrange Grassmannians. C.R.Acad. Sci. Paris, Ser. I, Vol. 345, Issue 11 (2007) 647–652.

[18] C. Li, I. Zelenko: Differential geometry of curves in Lagrange Grassmannianswith given Young diagram, Differ. Geom. Appl., 27 (2009) 723-742.

[19] C.Li, I. Zelenko: Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries, J. Geom. Phys. 61 (2011) 781-807.


[20] J. Lott, C. Villani: Ricci curvature for metric-measure spaces via optimaltransport, Ann. of Math. (2), in press

[21] J. Lott, C. Villani: Weak curvature conditions and functional inequalities. J.Funct. Anal. 245 (2007), no. 1, 311–333

[22] R. Montgomery: A tour of subriemannian geometries, their geodesics andapplications. Mathematical Surveys and Monographs, 91. American Mathe-matical Society, Providence, RI, 2002

[23] S. Ohta: On the measure contraction property of metric measure spaces.Comment. Math. Helv. 82 (2007), no. 4, 805–828

[24] S. Ohta: Finsler interpolation inequalities, to appear in Calc. Var. PartialDifferential Equations

[25] H.L. Royden: Comparison theorems for the matrix Riccati equation. Comm.Pure Appl. Math. 41 (1988), no. 5, 739–746.

[26] K.T. Sturm: On the geometry of metric measure spaces. Acta Math. 196,no.1, 65-131 (2006)

[27] K.T. Sturm: On the geometry of metric measure spaces II. Acta Math. 196,no. 1, 133-177 (2006)

[28] N. Tanaka: A differential geometric study on strongly pseudo-convex mani-fold, Kinokunya Book Store Co., Ltd., Kyoto, 1975.

[29] S. Tanno: Variational problems on contact Riemannian manifolds, Trans.Amer. Math. Soc. 314 n. 1 (1989), 349379.

[30] S.M. Webster: Pseudo-Hermitian structures on a real hypersurface. J. Differ-ential Geometry, 13, 25-41 (1978).

Room 216, Lady Shaw Building, The Chinese University of HongKong, Shatin, Hong Kong

E-mail address: [email protected]

Department of Mathematics, Tianjin University, Tianjin, 300072,P.R.China


Department of Mathematics, Texas A&M University, College Sta-tion, TX 77843-3368, USA


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